Dear Ed,
 
the smallest number whose sum of divisors is 10 times the number is:
 
 487# * 29# * 10! / 2
 
which is about 1.19630286446804*10^217 and its factor is 10.00036
 
(Lejeunes number is about: 2.29353535102896E+222 and its factor is 10.0129)
 
I have also found the samllest numbers which exceed the factor 1,2,3,4,5,6,7,8,9,10. 
This gives the sequence A023199 (which - alas - is already known)
 
I enjoyed also your mentioning of DROD, which I did not know so far. I have bought it 
and am now trying to solve some of its riddles - thats really a labyrinth of dungeons....
 
Thanks
 
Yours - Bodo Zinser from Munich

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I stop now, this one is the smallest I found (and I supposed to be the
smallest at all)
1196302864468037060143273209658450423930161241757076738827918396950538234505
9016648535347468836046455234043935194500296748473838972491647043636698551317
157043820830462348056767777839861694764691622417511967632039040000
 = 487# * 29# * 10! / 2 (=10.0004 times sum of it's divisor)

May be you can wrote about it on your page http://www.mathpuzzle.com/

Thanks,
Vincent Lejeune
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Smallest n : sigma0(n) / n >= 11 that I found was
2^10 * 3^6 * 5^4 * 7^3 * ( 11 ... 29 )^2 * (31 ... 487) or
487# 29# 10! / 2
 
Regards,
 
Brian Trial 
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